March 21, 2022

Competition ….

\[{dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2 \over K_1}\right)\]

\[{dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta {N_1 \over K_2}\right)\]

is the outcome of 2 density dependent phenomena:

  1. Intra-specific competition (\(K_1\) and \(K_2\))
  2. Inter-specific competition (\(\alpha\) and \(\beta\))
Patterns depend on the scale and particulars of the competition intensity.

Focus on the the isoclines of zero growth

\[N_2 = {K_1 \over \alpha} - {1\over \alpha} N_1\]

\[N_2 = K_2 - \beta N_1\]

Extinction vs. Equilibria

If:

  • \({1\over\beta} < {K_1 \over K_2} > \alpha\); \(N_1\) wins
  • \({1\over\beta} > {K_1 \over K_2} < \alpha\); \(N_2\) wins
  • \({1\over\beta} < {K_1 \over K_2} < \alpha\); \(N_1\) or \(N_2\) wins
  • \({1\over\beta} > {K_1 \over K_2} > \alpha\); \(N_1\) and \(N_2\) coexist

Equilibria (stable or not) at:

  • \(N^*_1 = {K_1 - \alpha K_2 \over 1-\alpha\beta}\)
  • \(N^*_2 = {K_2 - \beta K_1 \over 1-\alpha\beta}\)

Co-existence essentially occurs when ….

Intra-specific > Inter-specific competition

\[{1\over\beta} > {K_1 \over K_2} > \alpha\]

\(\beta\) and \(\alpha\) small, in a very particular way relative to the ratio of carrying capacities.

Detecting and quantifying competition

Is Very Difficult!

Gerbil competition experiment

Gerbil coexistence and habitat partitioning

  • Zone I, both species prefer the semistabilized dune.
  • Zone II G. pyramidum prefers the semistabilized dune and G. allenbyi exhibits random utilization of both habitats
  • Zone III G. allenbyi exhibits preference for the stabilized sand and G. pyramidum prefers semistabilized dune.
  • Zone IV G. allenbyi prefers stabilized sand and G. pyramidum selects semistabilized dune but also uses the stabilized sand.

Ultimately ….

Gerbils CAN get along. But the isoclines are compilated!

Predation

Predation

an ecological process where one organism (the predator) consumes another (the prey).

  • Provides most of the principle route of energy flow through ecosystems
  • Strong selective pressure
  • Chief source of density dependent effects in regulation of many animal (and plant) populations

Predation is any transfer of energy up a trophic level!

  • Herbivory: animals eat plants
    • granivores: eat grain
    • frugivores: eat fruit
  • Parasitism: animals eat other animals without (immediately) killing them
  • Carnivory: animals eat other animals, killing them
  • Cannibalism: animals eat their conspecifics

Basic principles (to model)

  • Growth in predator population depends on the number of prey
  • Growth in prey population depends on the number of predators
  • As an extension of competition
  • An extreme form of competition

Intra-guild predation

Start with competition …

\[{dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right)\]

\[{dV \over dt} = r_v V \left(1-{V \over K_v} - \beta { V\over K_v}\right)\]

And ADD some extra terms

Species 1 (competitor AND predator \(P\)) benefits:

\[{dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right) + \gamma VP\]

Species 2 (competitor AND prey \(V\)) suffers:

\[{dV \over dt} = r_v V \left(1-{V \over K_v} - \beta { V\over K_v}\right) - \sigma V P\]

Some Definitions

\[{dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right) + \gamma VP\]

\[{dV \over dt} = r_v V \left(1-{V \over K_v} - \beta { V\over K_v}\right) - \sigma V P\]

  • \(P\) = predators
  • \(V\) = prey [victims]
  • \(\gamma\) (gamma) = conversion factor - how much does predators population benefit from eating prey?
  • \(\sigma\) (sigma) = capture efficiency - how much does prey population suffer from getting eaten?

Inspect the isoclines!

\[P^* = K_p - \left(\alpha - {\gamma K_p \over r_p}\right) V\]

\[V^* = K_v - \left(\beta + {\sigma K_v \over r_v}\right) P\]

Qualitatively

\(P\) isocline slope becomes STEEPER (and y - intercept increases). \(V\) isocline also becomes STEEPER (and x-intercept decreases)

Rotating isocline

Equilibrium shifts UP toward the predator \(K_1\) and DOWN from prey \(K_2\), as you’d expected.

The Lotka-Volterra Predator-Prey Model

That was complicated! Let’s simplify RADICALLY and eliminate ALL competition and density dependenec (\(\alpha = \beta = 0\), \(K_v = K_p = \infty\)).

\[{dP \over dt} = -q P + \gamma VP\] \[{dV \over dt} = r V - \sigma VP\]

Predators are always dying at rate \(q\), BUT grow in propotion to prey.

Prey grow exponentially at rate \(r\), BUT die off in proportion to predator.

Assumptions

Lots! And mainly unrealistic!

  • Standard continuous modeling stuff:
    • No age structure
    • Closed population
    • No time lag
  • Instantaneous B & D responses of P to V and V to P
  • Mass action: perfect population mixing in proportion to number of individuals

BUT it’s still a fun model to play with - and a good starting point.

Prey dynamics

\[{dV \over dt} = r V - \sigma VP\]

\(V\) - prey; \(P\) - predators; \(r\) - growth rate; \(\sigma\) - “capture efficiency”

No density dependence \(K\), No competition (\(\alpha N_2\)), only removal by \(P\) at rate \(\sigma\).

  • Large \(\sigma\) - strong effect: each moose killed by each wolf has big impact
  • Small \(\sigma\) - weak effect: bats and mosquitoes … hard to make an impact.

Predator dynamics

\[{dP \over dt} = - qP + \gamma VP \]

\(P\) - predators; \(V\) - prey; \(q\) - mortality rate; \(\gamma\) - “conversion” efficiency, e.g. how many prey = new predators

  • High \(\gamma\): high dependency on each discrete prey item, e.g. anaconda and capybara
  • Low \(\gamma\): low dependency on each discrete prey item, e.g. blue whale and krill

Obtain equilibria

\[\widehat{P} = {r\over \sigma}\]

  • Number of predators associated with 0 growth in prey population
  • ratio of prey growth to predator capture efficiency
  • ratio of inputs to outputs

\[\widehat{V} = {q\over \gamma}\]

  • Number of prey associated with 0 growth in predator population
  • ratio of predator death rate to efficiency with which prey becomes predator growth
  • ratio of outputs to inputs

Isoclines!

Prey Isoclines

Predator Isoclines

Graphical analysis

Check out the numerical tool!

How does this look?

  • Persistent oscillations with no convergence
  • (1/4 cycle out of phase)
    • \(P_{max}\) and \(P_{min}\) occur at alternating mean \(V\)
    • \(V_{max}\) and \(V_{min}\) occur at alternating mean \(P\)

2 exceptions - equilibrium - extreme starting point

Hare-Lynx dataset